Just-in-Time-Teaching and other gadgets. Richard Cramer-Benjamin Niagara University

Transcription

1 Just-in-Time-Teaching and other gadgets Richard Cramer-Benjamin Niagara University

2 The Class MAT 443 Euclidean Geometry 26 Students 12 Secondary Ed (9-12 or 5-12 Certification) 14 Elementary Ed (1-6, B-6, or 1-9 Certification)

3 The Class Venema, G., Foundations of Geometry, Preliminaries/Discrete Geometry Axioms of Plane Geometry Neutral Geometry Euclidean Geometry Circles Transformational Geometry 2 weeks 3 weeks 3 weeks 3 weeks 1 week 2 weeks

4 Other Sources Requiring Student Questions on the Text Bonnie Gold How I (Finally) Got My Calculus I Students to Read the Text Tommy Ratliff MAA Inovative Teaching Exchange

5 JiTT Just-in-Time-Teaching Warm-Ups Physlets Puzzles On-line Homework Interactive Lessons JiTTDLWiki JiTTDLWiki

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7 Goals Teach Students to read a textbook Math classes have taught students not to read the text. Get students thinking about the material Identify potential difficulties Spend less time lecturing

8 Example Questions For February 1 Subject line WarmUp 3 LastName Due 8:00 pm, Tuesday, January 31. Read Sections Be sure to understand The different axiomatic systems (Hilbert's, Birkhoff's, SMSG, and UCSMP), undefined terms, Existence Postulate, plane, Incidence Postulate, lie on, parallel, the ruler postulate, between, segment, ray, length, congruent, Theorem 5.4.6*, Corrollary 5.4.7*, Euclidean Metric, Taxicab Metric, Coordinate functions on Euclidean and taxicab metrics, the rational plane. Questions Compare Hilbert's axioms with the UCSMP axioms in the appendix. What are some observations you can make? What is a coordinate function? What does it have to do with the ruler placement postulate? What does the rational plane model demonstrate? List 3 statements about the reading. A statement can be something significant you learned or a question you have. At least two of the statements should be questions.

9 One point geometry contains one point and no lines. Which Incidence Axioms does one point geometry satisfy? Justify your answer. I don't think it satisfies any of the incidence axioms because there aren't any lines; all of the incidence axioms involved lines and multiple points. One point geometry does not satisfy any of the incidence axioms. Incidence axiom 1 says "for every pair of distinct points", but one-point geometry only has one point, not a pair of points. Incidence axiom 2 says "for every line l", but one point geometry does not have any lines. Incidence axiom 3 says "there exist three points", but in one point geometry there do not exist three points. Therefore, it doesn't satisfy any of the incidence axioms.

10 One point geometry satisfies Incidence Axiom 1 (For every pair of distinct points P and Q there exists exactly one line L such that both P and Q lie on L) and Incidence Axiom 2 (For every line L there exist at least two distinct points P and Q such that both P and Q lie on L). This can be illustrated by using If /Then statements. For example in Incidence Axiom 1 the If /Then statement would say: If for every pair of distinct points P and Q Then there exists exactly one line L such that both P and Q lie on L. Since both parts of this statement are false for one point geometry, a false/false statement results in a true statement which means that one point geometry does in fact satisfy Incidence Axiom 1. For Incidence Axiom 2, the If /Then statement would say: If for every line L there exist at least two distinct points P and Q then both P and Q lie on L. Again, both parts of this statement are false for one point geometry and again a false/false statement results in a true statement which means that one point geometry does satisfy Incidence Axiom 2. However, when it comes to Incidence Axiom 3 (There exist three points that do not all lie on any one line) it is not possible in one point geometry because in this cause there are 3 points.

11 Compare Hilbert's axioms with the UCSMP axioms in the appendix. What are some observations you can make? There are more undefined terms in Hilbert's axioms than in the UCSMP axioms. Distance and angle measure are an integral part of the UCSMP axioms and are not utilized in Hilbert's axioms. Additionally, UCSMP's axioms deal with area and volume while Hilbert's do not. Both axioms are organized by categories. Hilbert's axioms fall under five categories of axioms: Incidence, Order, Congruence, Parallels, and Continuity. UCSMP's axioms fall under eight categories of postulates: Point-Line-Plane, Distance, Triangle Inequality, Angle Measure, Corresponding Angle, Reflection, Area, and Volume. (Notice Hilbert's axioms are called "axioms" while UCSMP's are called "postulates.") Hilbert's axioms seem to use spatial relationships of angles, lines, and points. UCSMP's seem to branch off into many perspectives other than spatial.

12 What does the rational plane model demonstrate? The rational plane model demonstrates that every rational number is also a real number. The rational plane model demonstrates the existence of rational numbers through ordered pairs. Not only does the rational plane demonstrate rational numbers but it also demonstrates real numbers since all rational numbers are part of the set of real numbers. Although, the rational plane model demonstrates that the rational plane satisfies all five of Euclid's postulates, it also shows how the proof of Euclid's very first proposition breaks down in the rational plane. This means that the Euclid was using unstated hypotheses in his proofs, which also means that there is a gap in Euclid's proofs.

13 Questions Every Warm Up assignment will have one area for Give three questions or important things you learned from the reading. Questions must be stated as questions. Stating I didn t understand is not a question. Also, your question should not be something that can easily be looked up in the index.

14 What indications or hints were Euclid given in order to know the importance of his postulates and the remarkable role they would play in geometry? How did he pick and choose the axioms as well? I was under the impression that a model satisfies a postulate or a proof. The book states that there are two models used to show Euclid?s fifth postulate yet one of them satisfies it and the other model does not sitisfy it. How is then a model for Euclid?s fifth postulate?

15 In the section involving indirect proof it seems as if they tell you the exact manner in which to do a RAA and then show cases in which it is considered sloppy when it seems as if they are following the steps listed above in the description of the method of an indirect proof, why is this considered sloppy? Are certain elements of geometry more worthy or do they lend themselves better to a RAA? How do we know when to use an RAA, when is it most effective?

16 When we are writing proofs in class, should we try and use all of the six kinds of reason? Would they all apply at once or would that rarely be the case? Why is it not enough to provide experimental evidence that a theorm is true? When writing a proof, how would one know to write a proof using the logic formula, induction, proof by contradiction, etc.? what are some hints to tell when to use each proof process?

17 Loud Thinking H. A. Peelle, Alternative Modes for Teaching Mathematical Problem Solving: An Overview, The Journal of Mathematics and Science: Collaborative Explorations 4 (Spring 2001)

18 Loud Thinking Students work in pairs. Solver works on the problem Should verbalize thoughts while trying to solve problem. Recorder records solving process Can give hints and read back previous attempts. Should not be paired problem solving

19 Loud Thinking Recorders found it difficult not to take part in the problem solving Students liked that it forced them to be aware of the problem solving process. Many students are afraid to write down or say something incorrect. This is counter to what we think of as good problem solving. Weak students got to observe strong problem solvers.

20 Collaborative Oral Take Home Exam MAA Notes #49 - Annalisa Crannell Students work in groups of 3 or challenging problems 1 week to solve the problems Each group schedules a half-hour exam time Each member selects a problem at random and solves the problem at the board with no notes. Group is graded on written solutions, individuals are graded on oral solutions

21 Collaborative Oral Take Home Exam Originally made the grade 100% oral. Too much pressure for students Most groups did very well. Some difficulty with student schedules. Nice way to make collaborative learning part of a higher stakes assessment.

22 Reading Out Loud Students pair up Take turns reading a sentence from the text. The listener then asks a question about the passage just read. Students read very quickly and have trouble asking probing questions.

23 Circle Time Groups of three Each person in the group gets a unique problem to work outside of class. Students then exchange solutions and evaluate groups members solutions

24 Circle Time Students have very little ability to evaluate the correctness of a proof. Students can not follow the logical arguments of another student Students can comment on notation

25 Web pages My Home Page JiTT n_page MAA Inovative Teaching Exchange ge.html

26 Requirements Secondary Calc I, II, and III Foundations Linear Algebra Abstract Algebra History of Math Prob and Stat I Math Modeling Euclidean Geometry Senior Seminar Elementary Intro to Statistics Elem. Ed Math I and II Calc I and II Foundations Linear Algebra History of Math Math Modeling Euclidean Geometry Senior Seminar